# Research Highlights

Work supported in part by the National Science Foundation NSF CAREER Award DMS#1554733 and the UC Davis Chancellor's Fellowship

My research is primarily in harmonic analysis, with special focus on problems related to Whitney extensions and their applications. I have also done some work on partial differential equations (specifically, the capillary equation for compressible fluids, the quasi-geostrophic equation, Buckley-Leverett Equation, and semi-linear wave equations).

### Whitney Extensions

$$C^{m,1}(\mathbb{R}^n)$$ Extensions of Bounded Depths

At the basic level, the extension problems ask for efficient methods to find a smooth function to fit through some data points while minimizing some quantities (e.g. norms, energy functionals).

Given $$E \subset \mathbb{R}^n$$ (arbitrary), $$f:E \rightarrow \mathbb{R}$$ and a function space equipped with a norm $$\|\cdot\|$$ (which can be $$C^{m}(\mathbb{R}^n)$$,$$C^{m,1}(\mathbb{R}^n)$$, $$W^{m,p}(\mathbb{R}^n)$$ (Sobolev spaces)), we would like to answer the following questions:

• How can we compute the smallest $$\|F\|$$ such that $$F = f$$ on $$E$$ ? We will often be happy with computing the least $$\|F\|$$ up to a multiplicative constant $$C$$ depending only on $$m$$ and $$n$$; an $$F$$ that achieves the least possible norm up to a multiplicative constant $$C(m,n)$$ is called an almost-optimal interpolant.
• Can we construct an almost-optimal interpolant $$F$$ linearly? That is, can we take $$F$$ in the form $F(x) = \sum_{y \in E} a(x,y) f(y)?$ Here $$a(x,y)$$ is independent of $$f$$.
• If $$E$$ is a finite set with $$N$$ points, can we produce an almost optimal interpolant F efficiently? For example, using $$C(m,n)N\log N$$ computer operations.

For $$C^{m,1}(\mathbb{R}^n)$$(the space of m-times continuously differentiable functions whose highest derivatives are Lipschitz continuous), in Adv. Math. 2010, I showed that not only can the extension be constructed linearly but also it can be taken to have a sparse form: That is, the value of the extension (as well as its derivatives) at any point $$x \in \mathbb{R}^n$$ is a linear combination of boundedly many given function values: More precisely, given any set $$E \subset \mathbb{R}^n$$ there exists an operator $$T$$ such that for all $$x$$ and $$f: E \rightarrow \mathbb{R}$$, we have

$Tf(x) = \sum_{y \in S(x)} a(x,y) f(y),$

where $$S(x)$$ is a subset of $$E$$ with almost $$C(n,m)$$ points.

For example, in one dimension, the function's value (as well as derivatives) of the best $$C^{m,1}(\mathbb{R})$$-interpolant at any query point $$x$$ depends on at most $$m+1$$ given (consecutive) data points in $$E$$; this is apparent using the divided-differences.

For finite sets $$E$$, Fefferman showed in 2009 that for $$C^{m,1}(\mathbb{R}^n)$$ and $$C^{m}(\mathbb{R}^n)$$ such an extension (linear and sparse) exists, but for infinite sets $$E$$, for $$C^{m}(\mathbb{R}^n)$$ there is a counterexample. Linearity for the extension (let alone with the sparse representation) is far from being obvious for infinite sets; the result I obtained in Adv. Math. 2010 is the first result for arbitrary sets.  The bounded depth operators (aka operators with sparse representation) provide the foundation for supplying an efficient algorithm for computing the best $$C^{m,1}(\mathbb{R}^n)$$ interpolant for arbitrary set $$E$$ of $$\mathbb{R}^n$$;

Sobolev Extensions

In the setting of Sobolev spaces, the situation is completely different from $$C^{m,1}(\mathbb{R}^n)$$. We showed that in general a Sobolev extension does not admit a sparse (aka "bounded depth") representation (joint work with Fefferman and Israel, Rev. Mat. Iberoam., 2014). For an arbitrary set $$E \subset \mathbb{R}^n$$, in J. Amer. Math. Soc., 2013), we showed that for the Sobolev spaces of functions whose m-th derivatives are in $$L^{p}(\mathbb{R}^n)$$ with $$n < p <\infty$$, there is a bounded linear extension operator. This is the first result of its kind without any underlying assumption on the $$E$$. (This result differs substantially and improves drastically previous results that rely on the assumptions $$E$$ is a nice set with good geometry, e.g., John domains); moreover, in the same paper J. Amer. Math. Soc., 2013, by introducing some assisted quantities, we are able to produce an almost-sparse representation for the Sobolev extensions and for finite sets $$E$$, we have a formula for computing the almost-optimal norm. In a series of three papers totaling over 300 pages (joint with Fefferman and Israel) (Part I, Part II, Part III, Rev. Mat. Iberoam., 2016), we made all the steps in the J. Amer. Math. Soc., 2013 paper algorithmically effective and obtained $$\mathcal{O}(N\log N)$$ algorithms as in Fefferman-Klartag for Sobolev spaces. Our results yield the first robust algorithms for solving relaxed Dirichlet type of problems (i.e. minimizing energy functionals subject to boundary constraints up to a multiplicative constant depending on dimension and smoothness only) on arbitrary finite sets.

Selection Problems

In Geom. Funct. Anal., 2016, together with my co-authors, we solved the Brudnyi-Shvartsman Conjecture (1994), which states the following: Suppose $$E \subset \mathbb{R}^n$$ and at each $$x\in E$$ we are given a convex set $$K(x) \subset \mathbb{R}^d$$. Then there exists a constant $$C(m,n,d)$$ depending only on $$m, n, d$$ such that if for each subset $$S \subset E$$ with at most $$C(m,n,d)$$ points there is a $$C^{m,1}(\mathbb{R}^n,\mathbb{R}^d)$$ function $$\vec{F}^S$$ with norm at most one such that $$\vec{F}^S(x) \in K(x)$$ for all $$x \in S$$. Then there is a $$C^{m,1}(\mathbb{R}^n,\mathbb{R}^d)$$ function $$\vec{F}$$ function with norm bounded by $$M(m,n,d)$$ such that $$\vec{F}(x)\in K(x)$$ for all $$x \in E$$.  In other words, from the effective local properties, we can deduce a global property. For $$d=1,2$$ and when $$m=0,1$$, the conjecture was settled by Brudnyi-Shvartsman in 1994 and Shvartsman (2001, 2002). If we assume further that $$K(x)$$ are symmetric about a point in $$\mathbb{R}^d$$ and in the conclusion we allow $$F(x) \in CK(x)$$ (i.e., $$K(x)$$ is allowed to be dilated by a factor) for some $$C$$ depending only on $$m,n,d$$, Fefferman proved the conjecture in Ann. Math. in 2007.

Nonnegative Interpolation

Given a finite subset $$E\subset \mathbb{R}^n$$, and a nonnegative function $$f: E \rightarrow [0,\infty)$$, we would like to answer the following questions:

1. Compute $\inf \{ \|F\|_{C^{m}(\mathbb{R}^n)}: F|_E=f, F\geq 0 \}$ up to a constant multiple that depends only on $$m$$ and $$n$$.

2. Compute a nonnegative function $$F$$ such that $$F = f$$ on $$E$$ and  $$\|F\|_{C^{m}(\mathbb{R}^n)}$$ is within a constant multiple (depending on $$m$$ and $$n$$) of the least possible.

We solved Problem 1 in a joint paper (Rev. Mat. Iberoam., 2017) by proving a Finiteness Principle.

In a series of 3 papers with my graduate student Fushuai (Black) Jiang, we consider the nonnegative interpolation for m = 2, n = 2 for finite sets E. We obtain the following results.

• In Paper #1  Nonnegative $$C^2(\mathbb{R}^2)$$ Interpolation (to appear in Advances in Mathematics, 2020), we improve the finiteness constant in the finiteness principle for nonnegative smooth interpolation obtained earlier in Rev. Mat. Iberoam., 2017 . We show that if one knows the norm of an optimal interpolant on all 64 point subsets of $$E$$, then one knows the (almost) optimal norm for any finite set $$E$$. Not proven sharp, but 64 is a substantial improvement on the previous known constant (Rev. Mat. Iberoam., 2017 yielded 5200 as a bound).
• In Paper #1  Nonnegative $$C^2(\mathbb{R}^2)$$ Interpolation (to appear in Advances in Mathematics, 2020), we also prove a sharp finiteness principle, which reads as follows: Given a finite set $$E\subset \mathbb{R}^2$$ with $$\#(E) = N$$, we can produce a list of subsets $$S_1,S_2,\cdots,S_L \subset E$$ such that $\begin{array}{l} L \leq 1000N, \\ \#(S_j) \leq 2000, \forall 1\leq j \leq L,\\ E = \bigcup_{1\leq j \leq L} S_j, \end{array}$ so that given any positive function $$f$$ on $$E$$, $\inf \{ \|F\|_{C^{m}(\mathbb{R}^n)}: F|_E=f, F\geq 0 \}$ can be obtained up to a universal multiplicative constant by $\max_{1\leq j \leq L}\inf \{ \|F\|_{C^{m}(\mathbb{R}^n)}: F|_{S_j}=f, F\geq 0 \}.$ (Here one imagines that $$\#(E) = N$$ is large, e.g., $$\gg 2000$$.)
• In Paper #1  Nonnegative $$C^2(\mathbb{R}^2)$$ Interpolation (to appear in Advances in Mathematics, 2020), we also show that contrary to the unconstrained problem, in general, a nonnegative $$C^2(\mathbb{R}^2)$$ extension operator is not linear. This result is first of its kind regarding the non-existence of linear extension operator.
• In Paper #2  C2(R2) Nonnegative Extension by Bounded-depth Operators (to appear in Advances in Mathematics, 2020), we extend the notion of bounded-depth (first defined for linear extension operators by C. Fefferman) to nonlinear extension operators, and we show that for finite $$E \subset \mathbb{R}^2, C^2_{+}(E)=$$(the space of $$C^2(\mathbb{R}^2)$$ nonnegative functions restricted to $$E$$) admits a nonnegative (nonlinear) extension operator of bounded-depth, which in a nutshell states that derivatives up to order 2 of the extension at any point $$x \in \mathbb{R}^2$$ can be obtained from sampling $$E$$ at a bounded number of points.
• The depth (see the bulletin point above) of an extension operator (both linear and nonlinear) measures the computational complexity of the extension. The existence of a linear extension operator of bounded depth is one of the main ingredients for the Fefferman-Klartag and Fefferman  algorithms for solving the interpolation problems without the nonnegative constraint. Equipped with a nonnegative (nonlinear) extension operator of bounded-depth (see the bulletin point above), in Paper #3  Algorithms for nonnegative $$C^2(\mathbb{R}^2)$$ interpolation (to appear), we provide algorithms solving the $$C^2(\mathbb{R}^2)$$ interpolation problems with nonnegative constraint.

Potential applications of these results are abundant: In probability, the density functions are required to be nonnegative. In physics, temperatures and energies are nonnegative quantities; given measurements on these qualities, we can determine the best fit that obeys the physical assumptions.

Applications to Algebra

The work on the extension problems resonates with various topics in pure and applied mathematics. J. Kollár is one of the first people to study the continuous closure of a polynomial ideal, i.e., the set of functions that can be written as a linear combination (with continuous functions as the coefficients) of the polynomials in an ideal. This problem lies at the intersection between analysis and real-algebraic geometry. As such it calls for tools from both disciplines. In a joint work (with Fefferman, Rev. Mat. Iberoam., 2013), we initiated the study of $$C^m, C^{m,\omega}$$ (i.e., differentiable) closures of polynomial ideals; we gave a necessary and sufficient condition to decide when a given function is in the $$C^m,C^{m,\omega}$$ closures of a given polynomial ideal by means of solving a vector-valued extension problem. In two recent papers, Solutions to a System of Equations for Cm Functions and Generators for the Cm-closures of Ideals (to appear in Rev. Mat. Iberoam., 2020) we showed how to classify all functions that are in the $$C^m,C^{m,\omega}$$ closures of a given polynomial ideal. Roughly speaking, we show that given a matrix $$A$$ of functions we can compute a finite list of linear differential operators such that $$AF = f$$ admits $$C^m$$ or $$C^{m,\omega}$$ solution $$F$$ if and only if $$f$$ is annihilated by the linear differential operators. Moreover, we can algorithmically compute the generators for all $$f$$ for which $$AF = f$$ admits a $$C^m$$ or $$C^{m,\omega}$$ solution $$F$$. This is akin to the classical Frobenius theorem, which allows one to find the maximal set of independent solutions to a system of first order linear PDEs. Our theorem allows higher-order differential operators.

### Partial Differential Equations

In On one-dimension semi-linear wave equations with null conditions (Adv. Math, 2018), jointly authored with Pin Yu and Shiwu Yang, we obtained a sharp result regarding the global behavior of solutions to the 1-dimensional semilinear wave equations with null conditions. The well-known celebrated results in two and three dimensions all rely on the following idea: The smallness of the initial data implies that the nonlinear equation can be solved for a sufficiently long time; a global solution can then be constructed once the nonlinearity decays sufficiently. This idea fails in one dimension as one-dimensional waves in general do not decay. Our approach is based on the following observation: If we think of the solution behaves as linear waves, then we may regard the wave as a combination of left- and right-traveling waves. For sufficiently long time, which is ensured by the smallness of the initial data, the left- and right-traveling waves will be separated in space. On the other hand, the null conditions can be phrased as left-traveling waves coupled only with right-traveling waves, since they are far away from each other for large time, the spatial decay now yields decay in time. This new decay mechanism is strongly in contrast with that in the higher dimensional cases, where the improved decay comes from the tangential derivative of the waves along outgoing light cones. In the proof, we introduced new weighted energy estimates.

In On the generalized Buckley-Leverett equation, (J. Math. Phys., 2016), (with Burczak and Granero-Belinchon), we study the generalized Buckley-Leverett equation with nonlocal regularizing terms. One of these regularizing terms is diffusive, while the other one is conservative. We prove that if the regularizing terms have order higher than one (combined), there exists a global strong solution for arbitrarily large initial data. In the case where the regularizing terms have combined order one, we prove the global existence of solution under some size restriction for the initial data. Moreover, in the case where the conservative regularizing term vanishes, regardless of the order of the diffusion and under a certain hypothesis on the initial data, we also prove the global existence of a strong solution, and we obtain some new entropy balances. Finally, we provide numerics suggesting that, if the order of the diffusion is less than 1, a finite time blow up of the solution is possible.

In The spine of an SQG almost-sharp front (Nonlinearity, 2012), (with C. Fefferman and J. Rodrigo), we considered a solution to the SQG equation in the form of 1 above a δ-neighborhood of a curve and -1 below the neighborhood and smoothly transitioning from 1 to -1 in the neighborhood. We showed that to such a solution we can associate a distinguished curve, which we called the spine, and that the spine satisfies the sharp-front equation modulo $$\mathcal{O}(\delta^2|\log \delta|)$$.

In On the capillary problems for compressible fluids (J. Math. Fluid Mech, 2007), (with R. Finn), we introduced a new capillary equation that accounted for compressibility in the fluids. We showed the well-posedness of solutions in a cylinder when the contact angle is less than $$\frac{\pi}{2}$$ and showed solutions do not in general exist when the contact angle is greater than $$\frac{\pi}{2}$$.

### Work in progress

1. Let $$A$$ be a matrix of functions and $$f$$ be a given function. (A) Decide when there exists a nonnegative function $$F \in C^m(\mathbb{R}^n),\mathbb{R}^d$$ (i.e., $$F=(F_1,\cdots,F_d)$$ with each $$F_i \in C^m(\mathbb{R}^n)$$) such that $$AF = f$$. (B) If such an $$F$$ exists, compute it. (C) Determine all $$f$$ for which $$AF = f$$ admits a nonnegative $$C^m$$ solution $$F$$. This is related to the Hilbert's 17th Problem, which asks to express a positive polynomial as a sum of squares of rational functions.

2. Selections for infinite set $$E$$, i.e., the full Brudnyi-Shvartsman Conjecture (1994), see discussion above.